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<span class="kw">let </span><font color="Maroon" title="c1">FCPS</font> be   <a href="anproj_2.html#NV10" title="ANPROJ_2:NV.10">up-3-dimensional</a>  <a href="anproj_2.html#NM1" title="ANPROJ_2:NM.1">CollProjectiveSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a9</font>, <font color="Olive" title="b3">b9</font>, <font color="Olive" title="b4">c9</font>, <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">p</font>, <font color="Olive" title="b9">q</font>, <font color="Olive" title="b10">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font>  st <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a9</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b9</font> &amp; <font color="Olive" title="b1">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c9</font> &amp; <font color="Olive" title="b5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">a9</font> &amp; <font color="Olive" title="b6">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b3">b9</font> &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font>,<font color="Olive" title="b7">c</font> <a href="projdes1.html#R2" title="PROJDES1:pred.2">constitute_a_quadrangle</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a</font>,<font color="Olive" title="b2">a9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b6">b</font>,<font color="Olive" title="b3">b9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">c</font>,<font color="Olive" title="b4">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font>,<font color="Olive" title="b8">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">a9</font>,<font color="Olive" title="b3">b9</font>,<font color="Olive" title="b8">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b6">b</font>,<font color="Olive" title="b7">c</font>,<font color="Olive" title="b9">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b3">b9</font>,<font color="Olive" title="b4">c9</font>,<font color="Olive" title="b9">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b5">a</font>,<font color="Olive" title="b7">c</font>,<font color="Olive" title="b10">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Olive" title="b2">a9</font>,<font color="Olive" title="b4">c9</font>,<font color="Olive" title="b10">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  holds <br/><font color="Olive" title="b8">p</font>,<font color="Olive" title="b9">q</font>,<font color="Olive" title="b10">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">o</font>, <font color="Maroon" title="c3">a9</font>, <font color="Maroon" title="c4">b9</font>, <font color="Maroon" title="c5">c9</font>, <font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</font>, <font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font>, <font color="Maroon" title="c11">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a9</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b9</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c9</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a9</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b9</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="projdes1.html#R2" title="PROJDES1:pred.2">constitute_a_quadrangle</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">a9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c4">b9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c5">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">b9</font>,<font color="Maroon" title="c9">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">b9</font>,<font color="Maroon" title="c5">c9</font>,<font color="Maroon" title="c10">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c5">c9</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  implies <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )</span><br/>





















<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a9</font>
 <span class="kw">and </span><br/><a NAME="E2:33"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b9</font>
 <span class="kw">and </span><br/><a NAME="E3:33"/><span class="lab"><font color="Green" title="E27">A3</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c9</font>
 <span class="kw">and </span><br/><a NAME="E4:33"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a9</font>
 <span class="kw">and </span><br/><a NAME="E5:33"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b9</font>
 <span class="kw">and </span><br/><a NAME="E6:33"/><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="projdes1.html#R2" title="PROJDES1:pred.2">constitute_a_quadrangle</a> 
 <span class="kw">and </span><br/><a NAME="E7:33"/><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">a9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E8:33"/><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c4">b9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E9:33"/><span class="lab"><font color="Green" title="E33">A9</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c5">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E10:33"/><span class="lab"><font color="Green" title="E34">A10</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c9">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E11:33"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">b9</font>,<font color="Maroon" title="c9">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E12:33"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E13:33"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c4">b9</font>,<font color="Maroon" title="c5">c9</font>,<font color="Maroon" title="c10">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E14:33"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E15:33"/><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c5">c9</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/>

<a NAME="E16:33"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="projdes1.html#D2" target="_self" title="PROJDES1:def.2">Def2</a></span>;<br/>
<a NAME="E17:33"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="projdes1.html#D2" target="_self" title="PROJDES1:def.2">Def2</a></span>;<br/>
<a NAME="E18:33"/><span class="lab"><font color="Green" title="E42">A18</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="projdes1.html#D2" target="_self" title="PROJDES1:def.2">Def2</a></span>;<br/>
<a NAME="E19:33"/><span class="lab"><font color="Green" title="E43">A19</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="projdes1.html#D2" target="_self" title="PROJDES1:def.2">Def2</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E44">A20</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c2">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E3:33_1"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font><span class="kw"> such that </span><br/><a NAME="E5:33_1"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c12">d</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="projdes1.html#T17" target="_self" title="PROJDES1:th.17">Th17</a></span>;<br/><a NAME="E6:33_1"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c12">d</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><a NAME="E7:33_1"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c8">c</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T18" target="_self" title="PROJDES1:th.18">Th18</a></span>;<br/><a NAME="E8:33_1"/><span class="lab"><font color="Green" title="E51">A27</font></span>: 
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c12">d</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="projdes1.html#T11" target="_self" title="PROJDES1:th.11">Th11</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">d9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font><span class="kw"> such that </span><br/><a NAME="E10:33_1"/><span class="lab"><font color="Green" title="E52">A28</font></span>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">d9</font>
 <span class="kw">and </span><br/><a NAME="E11:33_1"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
<font color="Maroon" title="c12">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">d9</font>
 <span class="kw">and </span><br/><a NAME="E12:33_1"/><span class="lab"><font color="Green" title="E54">A30</font></span>: 
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c13">d9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_2.html#D10" title="ANPROJ_2:def.10">ANPROJ_2:def 10</a></span>;<br/><a NAME="E13:33_1"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font><span class="kw"> such that </span><br/><a NAME="E15:33_1"/><span class="lab"><font color="Green" title="E55">A31</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c14">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E16:33_1"/><span class="lab"><font color="Green" title="E56">A32</font></span>: 
<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c13">d9</font>,<font color="Maroon" title="c14">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="anproj_2.html#D9" title="ANPROJ_2:def.9">ANPROJ_2:def 9</a></span>;<br/><a NAME="E17:33_1"/><span class="lab"><font color="Green" title="E57">A33</font></span>: 
<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c14">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:33_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E18:33_1"/>
<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c15">t</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font><span class="kw"> such that </span><br/><a NAME="E20:33_1"/><span class="lab"><font color="Green" title="E58">A34</font></span>: 
( <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c15">t</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  &amp; <font color="Maroon" title="c4">b9</font>,<font color="Maroon" title="c13">d9</font>,<font color="Maroon" title="c15">t</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="anproj_2.html#D9" title="ANPROJ_2:def.9">ANPROJ_2:def 9</a></span>;<br/><a NAME="E21:33_1"/>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">FCPS</font><span class="kw"> such that </span><br/><a NAME="E23:33_1"/><span class="lab"><font color="Green" title="E59">A35</font></span>: 
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c16">u</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">and </span><br/><a NAME="E24:33_1"/><span class="lab"><font color="Green" title="E60">A36</font></span>: 
<font color="Maroon" title="c5">c9</font>,<font color="Maroon" title="c13">d9</font>,<font color="Maroon" title="c16">u</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="anproj_2.html#D9" title="ANPROJ_2:def.9">ANPROJ_2:def 9</a></span>;<br/><a NAME="E25:33_1"/><span class="lab"><font color="Green" title="E61">A37</font></span>: 
<font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c14">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T18" target="_self" title="PROJDES1:th.18">Th18</a></span>;<br/><a NAME="E26:33_1"/>
not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c2">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E27:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A38</font></span>: 
not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c2">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:33_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="txt" href="#E3:33_1"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="projdes1.html#T22" target="_self" title="PROJDES1:th.22">Th22</a></span>;<br/><a NAME="E28:33_1"/><span class="kw">then </span>
not <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c5">c9</font>,<font color="Maroon" title="c13">d9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E10:33_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="projdes1.html#T23" target="_self" title="PROJDES1:th.23">Th23</a></span>;<br/><a NAME="E29:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">r</font>,<font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c14">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E14:33"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E15:33"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E15:33_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E16:33_1"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E23:33_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E24:33_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E27:33_1"><span class="lab"><font color="Green" title="E62">A38</font></span></a>, <a class="ref" href="projdes1.html#T21" target="_self" title="PROJDES1:th.21">Th21</a></span>;<br/><a NAME="E30:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A39</font></span>: 
<font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E31:33_1"/>
not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c2">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:33"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E32:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A40</font></span>: 
not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c2">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="projdes1.html#T22" target="_self" title="PROJDES1:th.22">Th22</a></span>;<br/><a NAME="E33:33_1"/><span class="kw">then </span>
not <font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T10" target="_self" title="PROJDES1:th.10">Th10</a></span>;<br/><a NAME="E34:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A41</font></span>: 
not <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c6">a</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E35:33_1"/><span class="lab"><font color="Green" title="E66">A42</font></span>: 
<font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c16">u</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T18" target="_self" title="PROJDES1:th.18">Th18</a></span>;<br/><a NAME="E36:33_1"/>
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c2">o</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:33"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E37:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A43</font></span>: 
not <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c12">d</font>,<font color="Maroon" title="c2">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:33_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="projdes1.html#T22" target="_self" title="PROJDES1:th.22">Th22</a></span>;<br/><a NAME="E38:33_1"/><span class="kw">then </span>
not <font color="Maroon" title="c4">b9</font>,<font color="Maroon" title="c5">c9</font>,<font color="Maroon" title="c13">d9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="txt" href="#E10:33_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="projdes1.html#T23" target="_self" title="PROJDES1:th.23">Th23</a></span>;<br/><a NAME="E39:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c15">t</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E12:33"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E13:33"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E20:33_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E23:33_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E24:33_1"><span class="lab"><font color="Green" title="E60">A36</font></span></a>, <a class="txt" href="#E37:33_1"><span class="lab"><font color="Green" title="E67">A43</font></span></a>, <a class="ref" href="projdes1.html#T21" target="_self" title="PROJDES1:th.21">Th21</a></span>;<br/><a NAME="E40:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A44</font></span>: 
<font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c10">q</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E41:33_1"/><span class="lab"><font color="Green" title="E69">A45</font></span>: 
<font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c15">t</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T18" target="_self" title="PROJDES1:th.18">Th18</a></span>;<br/><a NAME="E42:33_1"/>
<font color="Maroon" title="c13">d9</font>,<font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c14">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:33_1"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E43:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E10:33_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E34:33_1"><span class="lab"><font color="Green" title="E65">A41</font></span></a>, <a class="ref" href="projdes1.html#T8" target="_self" title="PROJDES1:th.8">Th8</a></span>;<br/><a NAME="E44:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A46</font></span>: 
not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c14">s</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E17:33_1"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="ref" href="projdes1.html#T19" target="_self" title="PROJDES1:th.19">Th19</a></span>;<br/><a NAME="E45:33_1"/><span class="lab"><font color="Green" title="E71">A47</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T18" target="_self" title="PROJDES1:th.18">Th18</a></span>;<br/><a NAME="E46:33_1"/>
not <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">b9</font>,<font color="Maroon" title="c13">d9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E10:33_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E32:33_1"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="ref" href="projdes1.html#T23" target="_self" title="PROJDES1:th.23">Th23</a></span>;<br/><a NAME="E47:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c14">s</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E11:33"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E15:33_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E16:33_1"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E20:33_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E32:33_1"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="ref" href="projdes1.html#T21" target="_self" title="PROJDES1:th.21">Th21</a></span>;<br/><a NAME="E48:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A48</font></span>: 
<font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c9">p</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T1" target="_self" title="PROJDES1:th.1">Th1</a></span>;<br/><a NAME="E49:33_1"/><span class="lab"><font color="Green" title="E73">A49</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c6">a</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="projdes1.html#T18" target="_self" title="PROJDES1:th.18">Th18</a></span>;<br/><a NAME="E50:33_1"/><span class="lab"><font color="Green" title="E74">A50</font></span>: 
not <font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c16">u</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E5:33_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E11:33_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="txt" href="#E12:33_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E15:33_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E16:33_1"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E20:33_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E23:33_1"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="txt" href="#E32:33_1"><span class="lab"><font color="Green" title="E64">A40</font></span></a>, <a class="ref" href="projdes1.html#T24" target="_self" title="PROJDES1:th.24">Th24</a></span>;<br/><a NAME="E51:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">u</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E52:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A51</font></span>: 
<font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c10">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:33_1"><span class="lab"><font color="Green" title="E68">A44</font></span></a>, <a class="txt" href="#E41:33_1"><span class="lab"><font color="Green" title="E69">A45</font></span></a>, <a class="txt" href="#E35:33_1"><span class="lab"><font color="Green" title="E66">A42</font></span></a>, <a class="ref" href="projdes1.html#T14" target="_self" title="PROJDES1:th.14">Th14</a></span>;<br/><a NAME="E53:33_1"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:33"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E54:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A52</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c11">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:33"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E49:33_1"><span class="lab"><font color="Green" title="E73">A49</font></span></a>, <a class="txt" href="#E7:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="projdes1.html#T14" target="_self" title="PROJDES1:th.14">Th14</a></span>;<br/><a NAME="E55:33_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:33"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E56:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A53</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c10">q</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:33"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E45:33_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="txt" href="#E7:33_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="ref" href="projdes1.html#T14" target="_self" title="PROJDES1:th.14">Th14</a></span>;<br/><a NAME="E57:33_1"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:33"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E58:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A54</font></span>: 
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c9">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E49:33_1"><span class="lab"><font color="Green" title="E73">A49</font></span></a>, <a class="txt" href="#E45:33_1"><span class="lab"><font color="Green" title="E71">A47</font></span></a>, <a class="ref" href="projdes1.html#T14" target="_self" title="PROJDES1:th.14">Th14</a></span>;<br/><a NAME="E59:33_1"/>
<font color="Maroon" title="c16">u</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:33_1"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E60:33_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A55</font></span>: 
<font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c11">r</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:33_1"><span class="lab"><font color="Green" title="E63">A39</font></span></a>, <a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="txt" href="#E35:33_1"><span class="lab"><font color="Green" title="E66">A42</font></span></a>, <a class="ref" href="projdes1.html#T14" target="_self" title="PROJDES1:th.14">Th14</a></span>;<br/><a NAME="E61:33_1"/>
<font color="Maroon" title="c14">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:33_1"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="ref" href="anproj_2.html#D7" title="ANPROJ_2:def.7">ANPROJ_2:def 7</a></span>;<br/><a NAME="E62:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">s</font>,<font color="Maroon" title="c15">t</font>,<font color="Maroon" title="c16">u</font>,<font color="Maroon" title="c9">p</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:33_1"><span class="lab"><font color="Green" title="E72">A48</font></span></a>, <a class="txt" href="#E25:33_1"><span class="lab"><font color="Green" title="E61">A37</font></span></a>, <a class="txt" href="#E41:33_1"><span class="lab"><font color="Green" title="E69">A45</font></span></a>, <a class="ref" href="projdes1.html#T14" target="_self" title="PROJDES1:th.14">Th14</a></span>;<br/><span class="kw">hence </span><a NAME="E63:33_1"/>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:33"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="txt" href="#E50:33_1"><span class="lab"><font color="Green" title="E74">A50</font></span></a>, <a class="txt" href="#E44:33_1"><span class="lab"><font color="Green" title="E70">A46</font></span></a>, <a class="txt" href="#E58:33_1"><span class="lab"><font color="Green" title="E78">A54</font></span></a>, <a class="txt" href="#E56:33_1"><span class="lab"><font color="Green" title="E77">A53</font></span></a>, <a class="txt" href="#E54:33_1"><span class="lab"><font color="Green" title="E76">A52</font></span></a>, <a class="txt" href="#E52:33_1"><span class="lab"><font color="Green" title="E75">A51</font></span></a>, <a class="txt" href="#E60:33_1"><span class="lab"><font color="Green" title="E79">A55</font></span></a>, <a class="ref" href="projdes1.html#T20" target="_self" title="PROJDES1:th.20">Th20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:33_2"/><span class="lab"><font color="Green" title="E45">A56</font></span>: 
not <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c2">o</font> <a href="projdes1.html#R1" title="PROJDES1:pred.1">are_coplanar</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> </span><br/><a NAME="E2:33_2"/><span class="kw">then </span>
not <font color="Maroon" title="c3">a9</font>,<font color="Maroon" title="c4">b9</font>,<font color="Maroon" title="c5">c9</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:33"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E8:33"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="txt" href="#E9:33"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="projdes1.html#T23" target="_self" title="PROJDES1:th.23">Th23</a></span>;<br/><span class="kw">hence </span><a NAME="E3:33_2"/>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="txt" href="#E7:33"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="txt" href="#E10:33"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="txt" href="#E11:33"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E12:33"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E13:33"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E14:33"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="txt" href="#E15:33"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E1:33_2"><span class="lab"><font color="Green" title="E45">A56</font></span></a>, <a class="ref" href="projdes1.html#T21" target="_self" title="PROJDES1:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E22:33"/>
<font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">r</font> <a href="collsp.html#R1" title="COLLSP:pred.1">is_collinear</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:33"><span class="lab"><font color="Green" title="E44">A20</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
